--- title: "Using a delay-adjusted case fatality ratio to estimate under-reporting" description: "Using a corrected case fatality ratio, we calculate estimates of the level of under-reporting for any country with greater than ten deaths" status: real-time-report rmarkdown_html_fragment: true update: 2020-08-25 authors: - id: tim_russell corresponding: true - id: joel_hellewell equal: 1 - id: sam_abbott equal: 1 - id: nick_golding - id: hamish_gibbs - id: chris_jarvis - id: kevin_vanzandvoort - id: ncov-group - id: stefan_flasche - id: roz_eggo - id: john_edmunds - id: adam_kucharski ---

Aim

To estimate the percentage of symptomatic COVID-19 cases reported in different countries using case fatality ratio estimates based on data from the ECDC, correcting for delays between confirmation-and-death.

Data availability

The under-reporting estimates for all countries can be downloaded as a single .csv file here.

Similarly, the prevalence estimates can be downloaded as a single .csv file here.

How to cite this work

If you wish to cite this work, please do cite the associated preprint [1]).

Methods Summary

The associated preprint[1], specifically the corresponding supplementary material contains a full description of the methods and limitations used to arrive at the estimates presented here.

Current estimates of under-reporting, prevalence and adjusted case curves along with reported cases

Temporal variation

Figure 1: Temporal variation in reporting rate. We calculate the percentage of symptomatic cases reported on each day a country has had more than ten deaths. We then fit a Gaussian Process (GP) to these data (see Temporal variation model fitting section for details), highlighting the temporal trend of each countries reporting rate. The red shaded region is the 95% CrI of fitted GP.

Prevalence estimates

Country Prevalence median (95% CrI) Total reported cases New reported cases (tallied over last 10 days) Population
Afghanistan 0.0095% (0.0045% - 0.023%) 36,326 1,067 38,928,341
Albania 0.19% (0.085% - 0.45%) 4,842 872 2,877,800
Algeria 0.034% (0.017% - 0.073%) 27,956 5,424 43,851,043
Angola 0.0069% (0.0031% - 0.017%) 836 245 32,866,268
Argentina 0.37% (0.2% - 0.7%) 162,482 43,225 45,195,777
Armenia 0.25% (0.13% - 0.5%) 37,339 2,752 2,963,234
Australia 0.063% (0.03% - 0.14%) 14,809 3,494 25,499,881
Austria 0.024% (0.014% - 0.05%) 20,448 1,042 9,006,400
Azerbaijan 0.073% (0.041% - 0.14%) 30,281 3,313 10,139,175
Bahamas 0.15% (0.078% - 0.38%) 334 244 393,248
Bahrain 0.41% (0.26% - 0.73%) 36,561 3,478 1,701,583
Bangladesh 0.03% (0.018% - 0.056%) 226,192 24,159 164,689,383
Belarus 0.061% (0.026% - 0.16%) 67,170 1,298 9,449,321
Belgium 0.056% (0.034% - 0.1%) 67,547 3,152 11,589,616
Benin 0.0037% (0.0017% - 0.0095%) 1,509 168 12,123,198
Bolivia 0.84% (0.45% - 1.6%) 71,792 13,673 11,673,029
Bosnia & Herzegovina 0.33% (0.16% - 0.73%) 10,468 2,334 3,280,815
Brazil 0.56% (0.31% - 1%) 2,442,362 367,515 212,559,409
Bulgaria 0.15% (0.071% - 0.35%) 10,529 1,983 6,948,445
Burkina Faso 0.00056% (0.00029% - 0.0019%) 1,080 48 20,903,278
Cameroon 0.0044% (0.0026% - 0.012%) 16,636 551 26,545,864
Canada 0.027% (0.016% - 0.053%) 114,552 4,939 37,742,157
Cape Verde 0.13% (0.071% - 0.34%) 1,631 314 555,988
Central African Republic 0.0051% (0.003% - 0.014%) 3,530 114 4,829,764
Chad 0.00057% (0.00025% - 0.0049%) 889 33 16,425,859
Chile 0.24% (0.12% - 0.88%) 347,722 19,077 19,116,209
China 0.00019% (0.00011% - 0.0012%) 86,724 1,300 1,439,323,774
Colombia 0.66% (0.37% - 1.2%) 257,056 66,401 50,882,884
Congo - Brazzaville 0.022% (0.013% - 0.046%) 3,057 567 5,518,092
Congo - Kinshasa 0.0017% (0.00073% - 0.0067%) 8,830 521 89,561,404
Costa Rica 0.28% (0.13% - 0.69%) 15,108 5,290 5,094,114
Côte d’Ivoire 0.013% (0.0083% - 0.024%) 15,332 1,743 26,378,275
Croatia 0.045% (0.02% - 0.13%) 4,675 646 4,105,268
Cuba 0.0017% (0.00096% - 0.0047%) 2,484 87 11,326,616
Cyprus 0.0044% (0.0024% - 0.011%) 914 23 1,207,361
Czechia 0.032% (0.019% - 0.06%) 15,302 1,661 10,708,982
Denmark 0.014% (0.0081% - 0.029%) 13,516 374 5,792,203
Djibouti 0.012% (0.0071% - 0.026%) 3,926 56 988,002
Dominican Republic 0.24% (0.15% - 0.49%) 64,151 12,637 10,847,904
Ecuador 0.17% (0.092% - 0.35%) 85,138 7,779 17,643,060
Egypt 0.032% (0.017% - 0.066%) 92,433 5,310 102,334,403
El Salvador 0.26% (0.13% - 0.58%) 14,871 3,527 6,486,201
Equatorial Guinea 0% (0% - 0%) 2,632 0 1,402,985
Estonia 0.0032% (0.0016% - 0.011%) 1,771 17 1,326,539
Eswatini 0.18% (0.083% - 0.45%) 1,826 587 1,160,164
Ethiopia 0.018% (0.0092% - 0.036%) 14,230 5,400 114,963,583
Finland 0.0035% (0.0018% - 0.013%) 7,131 80 5,540,718
France 0.027% (0.016% - 0.053%) 183,833 8,405 65,273,512
Gabon 0.08% (0.049% - 0.15%) 6,913 874 2,225,728
Georgia 0.0068% (0.0037% - 0.018%) 701 117 3,989,175
Germany 0.011% (0.007% - 0.019%) 206,113 4,668 83,783,945
Ghana 0.043% (0.026% - 0.075%) 33,246 6,564 31,072,945
Greece 0.0059% (0.0029% - 0.016%) 4,161 244 10,423,056
Guatemala 0.21% (0.11% - 0.42%) 44,966 7,011 17,915,567
Guinea 0.0088% (0.0054% - 0.017%) 6,367 564 13,132,792
Guinea-Bissau 0.0039% (0.0017% - 0.012%) 865 27 1,967,998
Guyana 0.033% (0.011% - 0.11%) 322 62 786,559
Haiti 0.024% (0.008% - 0.081%) 7,283 287 11,402,533
Honduras 0.27% (0.14% - 0.56%) 39,732 6,948 9,904,608
Hungary 0.0061% (0.0021% - 0.022%) 4,444 141 9,660,350
Iceland 0.011% (0.0059% - 0.028%) 292 16 341,250
India 0.059% (0.037% - 0.098%) 1,483,083 405,538 1,380,004,385
Indonesia 0.028% (0.015% - 0.054%) 100,301 15,421 273,523,621
Iran 0.27% (0.15% - 0.51%) 293,606 22,000 83,992,953
Iraq 0.22% (0.12% - 0.42%) 112,559 22,365 40,222,503
Ireland 0.015% (0.0059% - 0.041%) 25,822 142 4,937,796
Israel 0.35% (0.22% - 0.62%) 64,342 15,093 8,655,541
Italy 0.013% (0.0065% - 0.028%) 246,431 2,070 60,461,828
Jamaica 0.0071% (0.0033% - 0.029%) 319 79 2,961,161
Japan 0.0085% (0.0053% - 0.014%) 29,825 5,347 126,476,458
Jordan 0.0017% (0.00088% - 0.0048%) 278 72 10,203,140
Kazakhstan 0.16% (0.096% - 0.3%) 84,455 14,309 18,776,707
Kenya 0.024% (0.013% - 0.049%) 17,894 5,225 53,771,300
Kuwait 0.26% (0.16% - 0.46%) 63,636 5,475 4,270,563
Kyrgyzstan 0.24% (0.14% - 1.2%) 33,290 7,312 6,524,191
Latvia 0.0064% (0.0023% - 0.02%) 643 31 1,886,202
Lebanon 0.036% (0.02% - 0.082%) 3,783 1,107 6,825,442
Lesotho 0.037% (0.013% - 0.11%) 272 146 2,142,252
Liberia 0.0063% (0.002% - 0.042%) 1,119 79 5,057,677
Libya 0.069% (0.033% - 0.15%) 2,659 1,036 6,871,287
Lithuania 0.012% (0.0048% - 0.039%) 1,981 104 2,722,291
Luxembourg 0.3% (0.18% - 0.59%) 6,283 912 625,976
Madagascar 0.023% (0.013% - 0.047%) 9,078 2,841 27,691,019
Malawi 0.026% (0.012% - 0.06%) 3,271 802 19,129,955
Malaysia 0.00093% (0.00054% - 0.0026%) 8,805 140 32,365,998
Maldives 0.17% (0.1% - 0.33%) 1,232 439 540,542
Mali 0.00053% (0.00025% - 0.0019%) 2,495 41 20,250,834
Mauritania 0.032% (0.019% - 0.066%) 6,162 725 4,649,660
Mauritius 0% (0% - 0%) 8 0 1,271,767
Mexico 0.63% (0.35% - 1.2%) 395,463 56,576 128,932,753
Moldova 0.22% (0.11% - 0.45%) 23,074 2,360 4,033,963
Montenegro 0.57% (0.26% - 1.3%) 2,568 1,229 628,062
Morocco 0.045% (0.022% - 0.097%) 20,881 3,872 36,910,558
Mozambique 0.0019% (0.0011% - 0.0044%) 732 266 31,255,435
Namibia 0.058% (0.031% - 0.14%) 811 640 2,540,916
Nepal 0.0089% (0.0054% - 0.022%) 18,265 1,250 29,136,808
Netherlands 0.019% (0.011% - 0.035%) 53,144 1,570 17,134,873
New Zealand 0.00027% (1e-04% - 0.0011%) 435 4 4,822,233
Nicaragua 0.012% (0.0055% - 0.04%) 3,423 292 6,624,554
Niger 0.00034% (0.00014% - 0.0017%) 1,131 28 24,206,636
Nigeria 0.0053% (0.0031% - 0.01%) 41,083 5,073 206,139,587
North Macedonia 0.29% (0.14% - 0.63%) 10,172 1,205 2,083,380
Norway 0.0043% (0.0024% - 0.011%) 8,840 102 5,421,242
Oman 0.56% (0.28% - 2.3%) 76,574 11,554 5,106,622
Pakistan 0.011% (0.0066% - 0.021%) 275,195 11,729 220,892,331
Palestinian Territories 0.15% (0.095% - 0.27%) 12,756 3,870 5,101,416
Panama 0.72% (0.37% - 1.5%) 61,406 9,181 4,314,768
Paraguay 0.03% (0.016% - 0.08%) 4,374 919 7,132,530
Peru 0.26% (0.15% - 1.2%) 389,679 40,217 32,971,846
Philippines 0.031% (0.019% - 0.051%) 82,037 16,736 109,581,085
Poland 0.03% (0.015% - 0.065%) 43,380 3,656 37,846,605
Portugal 0.04% (0.023% - 0.13%) 50,439 1,909 10,196,707
Puerto Rico 0.31% (0.17% - 0.81%) 15,441 3,988 2,860,840
Qatar 0.28% (0.14% - 1%) 107,387 3,289 2,881,060
Romania 0.28% (0.15% - 0.56%) 45,877 9,211 19,237,682
Russia 0.11% (0.062% - 0.21%) 818,027 46,574 145,934,460
São Tomé & Príncipe 0.13% (0.07% - 0.34%) 653 122 219,161
Saudi Arabia 0.3% (0.15% - 0.61%) 268,801 20,518 34,813,867
Senegal 0.026% (0.012% - 0.06%) 9,387 1,095 16,743,930
Serbia 0.11% (0.055% - 0.27%) 24,095 3,643 8,737,370
Sierra Leone 0.0025% (0.0013% - 0.0087%) 1,733 82 7,976,985
Singapore 0.12% (0.068% - 0.25%) 49,959 3,183 5,850,343
Slovakia 0.0083% (0.0047% - 0.019%) 1,710 205 5,459,643
Slovenia 0.02% (0.0089% - 0.068%) 1,868 147 2,078,932
Somalia 0.00091% (0.00053% - 0.0024%) 3,170 67 15,893,219
South Africa 0.78% (0.43% - 1.5%) 452,255 101,650 59,308,690
South Korea 0.0019% (0.0011% - 0.0048%) 14,175 458 51,269,183
South Sudan 0.0027% (0.0013% - 0.0067%) 2,069 114 11,193,729
Spain 0.083% (0.05% - 0.27%) 279,864 18,527 46,754,783
Sri Lanka 0.00098% (0.00059% - 0.0019%) 1,936 101 21,413,250
Sudan 0.013% (0.0054% - 0.034%) 11,414 742 43,849,269
Suriname 0.19% (0.1% - 0.42%) 1,346 482 586,634
Sweden 0.048% (0.027% - 0.097%) 79,360 2,114 10,099,270
Switzerland 0.023% (0.014% - 0.044%) 34,372 984 8,654,618
Syria 0.0087% (0.003% - 0.024%) 487 178 17,500,657
Tajikistan 0.0087% (0.0053% - 0.02%) 7,235 401 9,537,642
Tanzania 0% (0% - 0%) 484 0 59,734,213
Thailand 0.00016% (8.6e-05% - 0.00044%) 3,120 48 69,799,978
Togo 0.0036% (0.0016% - 0.016%) 784 108 8,278,737
Tunisia 0.0021% (0.0011% - 0.0057%) 1,426 107 11,818,618
Turkey 0.023% (0.012% - 0.046%) 227,019 8,302 84,339,067
Ukraine 0.073% (0.037% - 0.15%) 65,651 7,545 43,733,759
United Arab Emirates 0.051% (0.031% - 0.11%) 59,024 2,466 9,890,400
United Kingdom 0.088% (0.047% - 0.17%) 300,592 5,413 67,886,004
United States 0.35% (0.22% - 0.59%) 4,290,248 578,799 331,002,647
Uruguay 0.018% (0.0069% - 0.051%) 795 158 3,473,727
Uzbekistan 0.031% (0.019% - 0.055%) 19,941 5,077 33,469,199
Venezuela 0.033% (0.02% - 0.06%) 15,687 4,505 28,435,943
Yemen 0.011% (0.0041% - 0.034%) 1,690 110 29,825,968
Zambia 0.11% (0.045% - 0.27%) 3,632 1,572 18,383,956
Zimbabwe 0.041% (0.02% - 0.09%) 2,153 1,226 14,862,927

Table 1: Estimates for the prevalence of COVID-19 in each country with greater than 10 deaths. We use the under-reporting estimates to adjust the reported case curves and tally these up over the last ten days as a proxy for prevalence. See Detailed Methods for more details.

Adjusted symptomatic case estimates

Figure 2: Estimated number of new symptomatic cases, calculated using our temporal under-reporting estimates. We adjust the reported case numbers each day - for each country with an under-reporting estimate - using our temporal under-reporting estimates to arrive at an estimate of the true number of symptomatic cases each day. The shaded blue region represents the 95% CrI, calcuated directly using the 95% CrI of the temporal under-reporting estimate.

Reported cases

Figure 3: Reported number of cases each day, pulled from the ECDC and plotted against time for comparison with our estimated true numbers of symptomatic cases each day, adjusted using our under-reporting estimates.

Current under-reporting estimates

Country Percentage of symptomatic cases reported (95% CI) Total cases Total deaths
Afghanistan 31% (21%-45%) 38,070 1,397
Albania 49% (35%-67%) 8,605 254
Algeria 78% (61%-96%) 41,858 1,446
Andorra 65% (27%-100%) 1,060 53
Angola 29% (21%-43%) 2,222 100
Argentina 51% (45%-57%) 350,854 7,366
Armenia 74% (58%-90%) 42,825 854
Australia 32% (24%-41%) 24,916 517
Austria 97% (84%-100%) 25,547 733
Azerbaijan 93% (81%-100%) 35,426 519
Bahamas 76% (43%-100%) 1,798 46
Bahrain 99% (92%-100%) 49,330 184
Bangladesh 89% (73%-100%) 297,083 3,983
Belarus 22% (13%-35%) 70,645 646
Belgium 97% (85%-100%) 81,998 9,996
Benin 82% (56%-100%) 2,115 39
Bolivia 34% (29%-39%) 110,148 4,578
Bosnia and Herzegovina 48% (36%-63%) 18,027 540
Brazil 65% (58%-72%) 3,622,861 115,309
Bulgaria 36% (27%-47%) 15,287 545
Burkina Faso 84% (43%-100%) 1,338 55
Cameroon 97% (72%-100%) 18,762 408
Canada 88% (65%-100%) 125,647 9,083
Cape Verde 92% (70%-100%) 3,532 37
Central African Republic 92% (48%-100%) 4,679 61
Chad 78% (21%-100%) 987 76
Chile 60% (31%-100%) 399,568 10,916
China 95% (28%-100%) 89,718 4,711
Colombia 46% (41%-52%) 551,696 17,612
Congo 49% (18%-99%) 3,850 77
Costa Rica 93% (78%-100%) 34,463 362
Cote dIvoire 98% (89%-100%) 17,506 114
Croatia 89% (66%-100%) 8,311 173
Cuba 94% (68%-100%) 3,717 91
Cyprus 88% (61%-100%) 1,460 21
Czechia 98% (88%-100%) 22,181 415
Democratic Republic of the Congo 36% (19%-59%) 9,841 250
Denmark 97% (87%-100%) 16,397 623
Djibouti 93% (72%-100%) 5,383 60
Dominican Republic 81% (60%-100%) 91,608 1,573
Ecuador 48% (39%-58%) 108,289 6,322
Egypt 25% (20%-31%) 97,478 5,280
El Salvador 63% (48%-79%) 24,811 678
Equatorial Guinea 92% (66%-100%) 4,926 83
Estonia 85% (46%-100%) 2,275 64
Eswatini 54% (39%-72%) 4,304 85
Ethiopia 57% (48%-68%) 42,143 692
Finland 88% (47%-100%) 7,938 335
France 96% (78%-100%) 244,854 30,528
Gabon 98% (90%-100%) 8,409 53
Gambia 27% (19%-41%) 2,585 87
Georgia 90% (64%-100%) 1,421 18
Germany 100% (97%-100%) 234,853 9,277
Ghana 99% (90%-100%) 43,622 263
Greece 87% (63%-100%) 8,819 242
Guatemala 39% (32%-47%) 68,533 2,611
Guinea 98% (89%-100%) 9,013 54
Guinea Bissau 64% (33%-97%) 2,149 33
Guyana 41% (24%-67%) 955 31
Haiti 60% (29%-99%) 8,110 196
Honduras 89% (71%-100%) 55,479 1,683
Hungary 46% (25%-78%) 5,191 613
Iceland 90% (62%-100%) 2,073 10
India 98% (83%-100%) 3,167,323 58,390
Indonesia 35% (30%-40%) 155,412 6,759
Iran 24% (21%-27%) 358,905 20,643
Iraq 62% (54%-71%) 207,985 6,519
Ireland 83% (59%-99%) 28,116 1,777
Israel 54% (41%-70%) 105,063 847
Italy 38% (30%-47%) 260,298 35,441
Jamaica 83% (51%-100%) 1,612 16
Japan 99% (92%-100%) 63,121 1,196
Jersey 31% (8.4%-89%) 364 32
Jordan 88% (57%-100%) 1,639 14
Kazakhstan 80% (68%-91%) 127,462 1,781
Kenya 83% (67%-97%) 32,557 554
Kosovo 21% (16%-27%) 12,448 467
Kuwait 99% (93%-100%) 80,960 518
Kyrgyzstan 97% (38%-100%) 43,245 1,501
Latvia 55% (30%-94%) 1,337 33
Lebanon 95% (80%-100%) 13,155 126
Lesotho 45% (27%-78%) 1,015 30
Liberia 55% (17%-100%) 1,290 82
Libya 62% (48%-78%) 11,009 199
Lithuania 76% (42%-100%) 2,673 85
Luxembourg 97% (82%-100%) 7,794 124
Madagascar 94% (81%-100%) 14,402 178
Malawi 64% (38%-90%) 5,419 169
Malaysia 95% (63%-100%) 9,274 125
Maldives 97% (88%-100%) 6,912 27
Mali 84% (44%-100%) 2,708 125
Mauritania 97% (74%-100%) 6,905 158
Mexico 17% (15%-19%) 563,705 60,800
Moldova 64% (53%-77%) 33,828 945
Montenegro 67% (48%-92%) 4,378 84
Morocco 50% (40%-60%) 53,252 920
Mozambique 95% (79%-100%) 3,440 21
Namibia 79% (56%-100%) 6,030 56
Nepal 65% (41%-100%) 32,678 157
Netherlands 99% (91%-100%) 67,062 6,193
New Zealand 68% (34%-100%) 1,339 22
Nicaragua 84% (43%-100%) 4,311 133
Niger 70% (22%-100%) 1,172 69
Nigeria 97% (88%-100%) 52,548 1,004
North Macedonia 54% (41%-70%) 13,595 564
Norway 90% (60%-100%) 10,323 264
Oman 74% (53%-93%) 84,509 637
Pakistan 97% (87%-100%) 293,711 6,255
Palestine 99% (94%-100%) 25,588 147
Panama 78% (65%-90%) 87,485 1,906
Paraguay 41% (30%-57%) 13,602 219
Peru 35% (30%-39%) 600,438 27,813
Philippines 99% (93%-100%) 194,252 3,010
Poland 78% (64%-94%) 62,310 1,960
Portugal 97% (71%-100%) 55,720 1,801
Puerto Rico 84% (59%-100%) 30,618 390
Qatar 91% (58%-100%) 117,266 194
Romania 39% (33%-46%) 79,330 3,309
Russia 61% (53%-68%) 961,493 16,448
San Marino 76% (16%-100%) 726 42
Sao Tome and Principe 88% (54%-100%) 892 15
Saudi Arabia 36% (30%-49%) 308,654 3,691
Senegal 61% (45%-81%) 13,013 272
Serbia 90% (67%-100%) 30,714 701
Sierra Leone 86% (43%-100%) 1,997 69
Singapore 93% (68%-100%) 56,404 27
Sint Maarten 71% (31%-100%) 408 17
Slovakia 94% (75%-100%) 3,424 33
Slovenia 62% (31%-98%) 2,665 128
Somalia 91% (51%-100%) 3,269 93
South Africa 49% (43%-56%) 611,450 13,159
South Korea 98% (85%-100%) 17,945 310
South Sudan 81% (55%-100%) 2,504 47
Sri Lanka 95% (70%-100%) 2,959 12
Sudan 35% (22%-55%) 12,836 815
Suriname 73% (50%-98%) 3,632 60
Sweden 98% (88%-100%) 86,721 5,813
Switzerland 99% (92%-100%) 39,959 1,720
Syria 31% (21%-45%) 2,293 92
Tajikistan 96% (67%-100%) 8,346 67
Thailand 85% (52%-100%) 3,402 58
Togo 74% (39%-100%) 1,295 27
Tunisia 52% (25%-95%) 2,893 71
Turkey 81% (67%-97%) 259,692 6,139
Uganda 45% (23%-85%) 2,362 22
Ukraine 69% (58%-81%) 106,754 2,293
United Arab Emirates 92% (52%-100%) 67,282 376
United Kingdom 97% (91%-100%) 326,614 41,433
United States of America 63% (53%-100%) 5,740,909 177,279
Uruguay 50% (30%-81%) 1,533 42
Uzbekistan 99% (93%-100%) 39,506 282
Venezuela 99% (93%-100%) 40,338 337
Vietnam 34% (20%-57%) 1,022 27
Yemen 7.2% (5.1%-10%) 1,916 555
Zambia 96% (78%-100%) 11,148 280
Zimbabwe 43% (33%-56%) 6,070 155

Table 2: Estimates for the proportion of symptomatic cases reported in different countries using cCFR estimates based on case and death timeseries data from the ECDC. Total cases and deaths in each country is also shown. Confidence intervals calculated using an exact binomial test with 95% significance.

Adjusting for outcome delay in CFR estimates

During an outbreak, the naive CFR (nCFR), i.e. the ratio of reported deaths date to reported cases to date, will underestimate the true CFR because the outcome (recovery or death) is not known for all cases [6]. We can therefore estimate the true denominator for the CFR (i.e. the number of cases with known outcomes) by accounting for the delay from confirmation-to-death [2].

We assumed the delay from confirmation-to-death followed the same distribution as estimated hospitalisation-to-death, based on data from the COVID-19 outbreak in Wuhan, China, between the 17th December 2019 and the 22th January 2020, accounting right-censoring in the data as a result of as-yet-unknown disease outcomes (Figure 1, panels A and B in [8]). The distribution used is a Lognormal fit, has a mean delay of 13 days and a standard deviation of 12.7 days [8].

To correct the CFR, we use the case and death incidence data to estimate the proportion of cases with known outcomes [2,7]:

\[ u_{t} = \frac{ \sum_{j = 0}^{t} c_{t-j} f_j}{c_t}, \]

where \(u_t\) represents the underestimation of the proportion of cases with known outcomes [2,6,7] and is used to scale the value of the cumulative number of cases in the denominator in the calculation of the cCFR, \(c_{t}\) is the daily case incidence at time, \(t\) and \(f_t\) is the proportion of cases with delay of \(t\) between confirmation and death.

Approximating the proportion of symptomatic cases reported

At this stage, raw estimates of the CFR of COVID-19 correcting for delay to outcome, but not under-reporting, have been calculated. These estimates range between 1% and 1.5% [2–4]. We assume a CFR of 1.4% (95% CrI: 1.2-1.7%), taken from a recent large study [4], as a baseline CFR. We use it to approximate the potential level of under-reporting in each country. Specifically, we perform the calculation \(\frac{1.4\%}{\text{cCFR}}\) of each country to estimate an approximate fraction of cases reported.

Temporal variation model fitting

We estimate the level of under-reporting on every day for each country that has had more than ten deaths. We then fit a Gaussian Process (GP) model using the library greta and greta.gp. The parameters we fit and their priors are the following: \[ \begin{aligned} &\sigma \sim \text{Log Normal(-1, 1)}: \quad &\text{Variance of the reporting kernel} \\ &\text{L} \sim \text{Log Normal(4, 0.5)}: \quad &\text{Lengthscale of the reporting kernel} \\ &\sigma_{\text{obs}} \sim \text{Truncated Normal(0, 0.5)}, \quad &\text{Variance of the obseration kernel, truncated at 0} \end{aligned} \] The kernel is split into two components: the reporting kernel \(R\), and the observation kernel \(O\). The reporting component has a standard squared-exponential form. For the observation component, we use an i.i.d. noise kernel to acccount for observation overdispersion, which can smooth out overly clumped death time-series. This is important as some countries have been known to report an unusually large number of deaths on a single day, due to past under-reporting.

In the sampling and fitting process, we calculate the expected number of deaths at each time-point, given the baseline CFR. We then use a Poisson likelihood, where the expected number of deaths is the rate of the Poisson likelihood, given the observed number of deaths

Approximating prevalence

We use the adjusted case curves, adjusted for under-reporting and for asymptomatic infections as a proxy for prevalence. Specifically, we tally up the adjusted new cases each day over the last ten days and calculate what percentage of the population in question this total equates to. This serves as a crude prevalence estimate. We assume ten days of infectiousness as taken from the mean of the total infectious period [9].

Adjusting case counts for under-reporting

We adjust the reported number of cases each day, pulled from the ECDC. Specifically, we divide the case numbers of each day by our “proportion of cases reported” estimates that we calculate each day for each country.*

Limitations

Implicit in assuming that the under-reporting is \(\frac{1.4\%}{\text{cCFR}}\) for a given country is that the deviation away from the assumed 1.4% CFR is entirely down to under-reporting. In reality, burden on healthcare system is a likely contributing factor to higher than 1.4% CFR estimates, along with many other country specific factors.

The following is a list of the other prominent assumptions made in our analysis:

Code and data availability

The code is publically available at https://github.com/thimotei/CFR_calculation. The data required for this analysis is a time-series for both cases and deaths, along with the corresponding delay distribution. We scrape this data from ECDC, using the NCoVUtils package [10].

The under-reporting estimates for all countries can be downloaded as a single .csv file here.

Similarly, global prevalence estimates can be downloaded as a single .csv file here

Acknowledgements

The authors, on behalf of the Centre for the Mathematical Modelling of Infectious Diseases (CMMID) COVID-19 working group, wish to thank DSTL for providing the High Performance Computing facilities and associated expertise that has enabled these models to be prepared, run and processed and in an appropriately-rapid and highly efficient manner.

References

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2 Russell TW, Hellewell J, Jarvis CI et al. Estimating the infection and case fatality ratio for covid-19 using age-adjusted data from the outbreak on the diamond princess cruise ship. medRxiv 2020.

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10 Abbott S MJ Hellewell J. NCoVUtils: Utility functions for the 2019-ncov outbreak. doi:105281/zenodo3635417 2020.